Linear stability of black holes with static scalar hair in full Horndeski theories: Generic instabilities and surviving models

In full Horndeski theories, we show that the static and spherically symmetric black hole (BH) solutions with a static scalar field φ whose kinetic term X is nonvanishing on the BH horizon are generically prone to ghost/Laplacian instabilities. We then search for asymptotically Minkowski hairy BH solutions with a vanishing X on the horizon free from ghost/Laplacian instabilities. We show that models with regular coupling functions of φ and X result in no-hair Schwarzschild BHs in general. On the other hand, the presence of a coupling between the scalar field and the Gauss-Bonnet (GB) term RGB2, even with the coexistence of other regular coupling functions, leads to the realization of asymptotically Minkowski hairy BH solutions without ghost/Laplacian instabilities. Finally, we find that hairy BH solutions in power-law F(RGB2) gravity are plagued by ghost instabilities. These results imply that the GB coupling of the form ζ(φ)RGB2 plays a prominent role for the existence of asymptotically Minkowski hairy BH solutions free from ghost/Laplacian instabilities.